41. CiteSeerX — Citation Query Proof Of The Double Bubble Conjecture CiteSeerX Scientific documents that cite the following paper Proof of the double bubble conjecture http://citeseerx.ist.psu.edu/showciting?cid=1528468

42. The Double Bubble Conjecture « OU Math Club Jan 26, 2010 This is the Double Bubble Conjecture. In 1995 it was proven by Hass, Hutchings, and Schlafy in the special case that both bubbles have the http://oumathclub.wordpress.com/2010/01/26/the-double-bubble-conjecture/

43. Historia Matematica Mailing List Archive: [HM] Double Bubble Co Research Announcement, February 25, 2000 Proof of the Double Bubble Conjecture by Michael Hutchings, Frank Morgan, Manuel Ritore, and Antonio Ros http://sunsite.utk.edu/math_archives/.http/hypermail/historia/mar00/0093.html

[HM] Double Bubble Conjecture Proved Subject: [HM] Double Bubble Conjecture Proved From: Antreas P. Hatzipolakis ( xpolakis@otenet.gr Date: Sat Mar 18 2000 - 12:21:54 EST Next message: Hans Lausch: "Re: [HM] Copernicus' Revolutionibi" Previous message: Antreas P. Hatzipolakis: "Re: [HM] Copernicus' Revolutionibi" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Research Announcement, February 25, 2000 Proof of the Double Bubble Conjecture by Michael Hutchings, Frank Morgan, Manuel Ritore, and Antonio Ros History. Archimedes and Zenodorus (see [K, p. 273]) claimed and Schwarz [S] proved that the round sphere is the least-perimeter way to enclose a given volume in R3. The Double Bubble Conjecture, long assumed true (see [P, pp. 300-301], [B, p. 120]) but only recently stated as a conjecture [F1, sect. 3], says that the familiar double soap bubble on the right in Figure 1, consisting of two spherical caps separated by a spherical cap or flat disc, meeting at 120-degree angles, provides the least-perimeter way to enclose and separate two given volumes. The analogous result in R2 was

44. Proof Of The Double Bubble Conjecture In Rn Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://www.springerlink.com/index/M42036623635477V.pdf

45. Proof Of The Double Bubble Conjecture In R4 And Certain Higher PACIFIC JOURNAL OF MATHEMATICS Vol. 208, No. 2,2003 PROOF OF THE DOUBLE BUBBLE CONJECTURE INR 4 AND CERTAIN HIGHER DIMENSIONAL CASES BenW. Reichardt, Cory Heilmann, Yuan Y. Lai, and http://pjm.math.berkeley.edu/pjm/2003/208-2/pjm-v208-n2-p09-p.pdf

46. Proof Of The Double Bubble Curvature Conjecture Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://www.springerlink.com/index/A1160637U5876501.pdf

47. Double Bubble Conjecture [Archive] - Physics Forums Archive Double Bubble Conjecture General Math Well, you can start off with Mathworld (http//mathworld.wolfram.com/DoubleBubble.html). http://www.physicsforums.com/archive/index.php/t-1086.html

48. Electronic Research Announcements The double bubble conjecture Author(s) Joel Hass; Michael Hutchings; Roger Schlafly http://www.ams.org/era/1995-01-03/S1079-6762-95-03001-0/home.html

49. THE STANDARD DOUBLE BUBBLE IS THE UNIQUE STABLE DOUBLE BUBBLE IN Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://www.ams.org/proc/2002-130-09/S0002-9939-02-06640-6/S0002-9939-02-06640-6.

50. Proof Of The Double Bubble Conjecture Outdated Archival Version. These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.kurims.kyoto-u.ac.jp/EMIS/journals/ERA-AMS/2000-01-006/2000-01-006.ht

Outdated Archival Version These pages are not updated anymore. They reflect the state of 20 August 2005 . For the current production of this journal, please refer to http://www.math.psu.edu/era/ This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/ Proof of the double bubble conjecture Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros Abstract. Retrieve entire article TeX source DVI PostScript Article Info ERA Amer. Math. Soc. (2000), pp. 45-49 Publisher Identifier: S 1079-6762(00)00079-2 Mathematics Subject Classification . Primary 53A10; Secondary 53C42 Key words and phrases . Double bubble, soap bubbles, isoperimetric problems, stability Received by the editors March 3, 2000 Posted on July 17, 2000 Communicated by Richard Schoen Comments (When Available) Michael Hutchings Department of Mathematics, Stanford University, Stanford, CA 94305 E-mail address: hutching@math.stanford.edu Frank Morgan Department of Mathematics, Williams College, Williamstown, MA 01267 E-mail address: Frank.Morgan@williams.edu

51. THE DOUBLE BUBBLE CONJECTURE File Format Adobe PostScript View as HTML http://emis.impa.br/EMIS/journals/ERA-AMS/1995-03-001/1995-03-001.ps

52. Double Bubble Conjecture Two partial Spheres with a separating boundary (which is planar for equal volumes) separate two volumes of air with less Area than any other boundary. http://mathserver.sdu.edu.cn/mathency/math/d/d385.htm

Double Bubble Conjecture Two partial Spheres with a separating boundary (which is planar for equal volumes) separate two volumes of air with less Area than any other boundary. The planar case was proved true for equal volumes by J. Hass and R. Schlafy in 1995 by reducing the problem to a set of 200,260 integrals which they carried out on an ordinary PC. See also Double Bubble References Haas, J.; Hutchings, M.; and Schlafy, R. ``The Double Bubble Conjecture.'' Electron. Res. Announc. Amer. Math. Soc. Eric W. Weisstein

53. Geometric Measure Theory And The Double Bubble Conjecture By Metin File Format PDF/Adobe Acrobat Quick View http://portal.ku.edu.tr/~agur/FINAL DRAFT.pdf

54. Nsf.gov - National Science Foundation (NSF) Discoveries - Double The team's solution of the Double Bubble Conjecture was announced in March 2000 before the Undergraduate Mathematics Conference at the RoseHulman Institute of Technology, and has http://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=100596&org=NSF

55. Geometric Measure Theory And The Double Bubble Conjecture File Format PDF/Adobe Acrobat Quick View http://portal.ku.edu.tr/~agur/SECOND DRAFT - BEAMER.pdf

56. Geometric Measure Theory And The Proof Of The Double Bubble Frank Morgan Geometric Measure Theory and the Proof of the Double Bubble Conjecture - Lecture 1 http://www.msri.org/publications/ln/msri/2001/minimal/morgan/1/index.html

57. Mathematical Recreations A notorious case is the Double Bubble Conjecture, which states that the shape formed when two bubbles coalesce consists of three spherical surfaces. http://www.fortunecity.com/emachines/e11/86/bubble.html

Web hosting Custom Email SiteBuilder Mathematical Recreations by Ian Stewart Double Bubble,Toil and Trouble The dodecahedron has 20 vertices, 30 edges and 12 faces- each with five sides. But what solid has 22.9 vertices, 34.14 edges and 13.39 faces -each with 5.103 sides? Some kind of elaborate fractal , perhaps? No, this solid is an ordinary, familiar shape, one that you can probably find in your own home. Look out for it when you drink a glass of cola or beer, take a shower or wash the dishes. I've cheated, of course. My bizarre solid can be found in the typical home in much the same manner that, say, 2.3 children can be found in the typical family. It exists only as an average. And it's not a solid; it's a bubble. Foam contains thousands of bubbles, crowded together like tiny, irregular polyhedra-and the average number of vertices, edges and faces in these polyhedra is 22.9, 34.14 and 13.39, respectively. If the average bubble did exist, it would be like a dodecahedron , only slightly more so. Coalescing Bubbles that enclose unequal volumes have shapes that remain a mathematical challenge.

58. Computer Images Of Double Bubbles By John Sullivan I created these images to illustrate the proof of the equalvolume case of the Double Bubble Conjecture by Hass and Schlafly in 1995. The bottom row shows a standard double bubble http://torus.math.uiuc.edu/jms/Images/double/

Standard and Nonstandard Double Bubbles John M Sullivan Click on any image for a larger (640x800) version. Please send me email ( jms@uiuc.edu ) for permission to publish these images, or to obtain even larger TIFF versions, with different background colors. These images show bubble clusters near equilibrium. The top row shows a standard double bubble of equal volumes, and a nonstandard cluster in which one bubble is a torus, forming a waist around the other. I created these images to illustrate the proof of the equal-volume case of the Double Bubble Conjecture by Hass and Schlafly in 1995. The bottom row shows a standard double bubble of unequal volumes (consisting of three spherical caps meeting at equal 120-degree angles), and a nonstandard bubble of the same volumes, in which the larger region is broken into two components (one a tiny ring around the other region). I created these images to illustrate the proof of the general Double Bubble Conjecture by Hutchings, Morgan, Ritore and Ros in 2000. In all four cases, the cluster is a surface of revolution. More details about the geometry of the examples with unequal volumes, including pictures of the generating curves, are available

59. Proof Of The Double Bubble Conjecture In Rn File Format PDF/Adobe Acrobat Quick View http://www.cs.uwaterloo.ca/~breic/talks/Proof of the Double Bubble Conjecture in

60. The Double Bubble Conjecture Outdated Archival Version. These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://emis.library.cornell.edu/journals/ERA-AMS/1995-03-001/1995-03-001.html

Outdated Archival Version These pages are not updated anymore. They reflect the state of 20 August 2005 . For the current production of this journal, please refer to http://www.math.psu.edu/era/ This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/ Comments on article The double bubble conjecture Joel Hass, Michael Hutchings, and Roger Schlafly Abstract. Retrieve entire article TeX source DVI PostScript Article Info ERA Amer. Math. Soc. (1995), pp. 98-102 Publisher Identifier: S1079-6762(95)03001-0 Mathematics Subject Classification . Primary 53A10; Secondary 49Q10, 49Q25. Key words and phrases . Double bubble; isoperimetric Received by the editors September 11, 1995 Communicated by Richard Schoen Comments Joel Hass Department of Mathematics, University of California, Davis, CA 95616 E-mail address: hass@math.ucdavis.edu Michael Hutchings Department of Mathematics, Harvard University, Cambridge, MA 02138 E-mail address: hutching@math.harvard.edu Roger Schlafly Real Software, PO Box 1680, Soquel, CA 95073

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